Question: Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7.
Solution: The units digit of $mn$ is $1^6 = 1$.  Searching for a units digit for $n$ (which is clearly odd), we find that $7 \cdot 3 = 1$, so $\boxed{3}$ is the units digit of $n$.